Not all the passes will be open mid May, but many will be. Of course there can always be a temporary closure, so do check before you drive. There are various routes from Zürich to Täsch. An obvious one is via the Brünig Pass (open all year round) and the car train between Kandersteg and Goppenstein (there's no road there). This route is pretty nice if you take your time and take a few local routes instead of just the main roads (like the Northern shore of Lake Brienz). The Susten Pass normally only opens up in June. That goes for the Furka Pass as well, which is another alternative in Summer.

From Täsch to Italy you will cross the Simplon Pass, which is also open all year long.

You don't need snow chains. They can only be used on a good layer of compact snow, which is not to be expected in May, and certainly not on the main roads. Should the road to Täsch be a problem you can park in Visp instead and take the train from there.